For higher-order \(C^0\)  finite elements, the lack of inter-element continuity is the cause of the different optical branches, and the final branch is due to the lack of first-order continuity (i.e., \(U^h\subset C^0 \setminus C^1\) ) [1]}. This is the reason for the superior spectral properties of the higher-order continuous spline basis functions used in isogeometric analysis [2]}. The mass-scaling term of bilinpenalty acts on this first-order inter-element continuity. It thus specifically targets the highest frequency branch. One potential approach for suppressing the frequencies from the lower optical branches is thus to add additional mass-scaling terms that involve jumps of higher-order derivatives. This concept is explored in an isogeometric analysis framework across patch-interfaces in [3]}, and on domain boundaries in [4]}. However, in the current article we aim to maintain the variational consistency of the finite element formulation, which only permits us to add a penalty on the natural conditions of bc2,trans2, i.e., the lowest derivative.
